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~P_\nu^{\mu + 1}(z)

~=

~
(z^2-1)^{-\frac{1}{2}} \{ (\nu - \mu) z P^\mu_\nu(z) - (\nu + \mu)P^\mu_{\nu - 1}(z)\}

Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)

NOTE: ~Q_\nu^\mu, as well as ~P_\nu^\mu, satisfies this same recurrence relation.

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